One day, I thougth "If every sequence is convergent, it is very nice." So I tried to define some equivalence class on the set of sequences. And I also tried to define operations such as addition, multiplication, etc, and the limit of the sequence of the "limit of the sequence". I also tried to extend the function to the "limit of the seauence".
But this is a very naive idea. And I could not go further with this idea.
Is there any work on this topic? That is, a work to define the limit for every sequence.
Please tell me any information.
Edit:
I try to clarify the idea. I would like to make a superset of numbers (let it denote S) which has some favorable properties. They may include the following:
(1) S is a vector space. If possible, a ring or a field.
(2) The sequences of S have a (linear?) operation called "limit". For number sequences, that operation is the same as the usual limit.
(3) If possible, the function of numbers can be extended to the function of S.
Let $X$ be a set. Equip $X$ with the trivial topology (consisting of only empty set and the set $X$). Every sequence in $X$ converges to each point in $X$. This is because if $x\in X$ and $(x_n)_n$ is a sequence in $X$, then the only open set containing $X$ must be $X$ itself! Obviously all the terms in the sequence are in $X$.